Testing sequence for boundaries and monotony

1. Jul 17, 2009

Juan Pablo

I'm trying to solve a problem from a practice test I was given. It says. Determine if the sequence is bounded and if it's decreasing or increasing:
$$a = \frac{arcsin(1/n)}{arctan(1/n)}$$

I don't really know where to start as I don't remember seeing how to test sequences for boundaries or anything like that and there's nothing on my book saying how to test it in the sequences chapter.

If anyone is interested, the practice test: http://profesores.usfq.edu.ec/valen/departamentomatematicas/files/exapractica/prac_destr_mat132/index.htm [Broken]

Spanish, sorry

Any hint?

Thanks!

Last edited by a moderator: May 4, 2017
2. Jul 17, 2009

Dick

The problem is really asking you about the behavior of the function f(x)=arcsin(1/x)/arctan(1/x) as x->infinity. The bounded part involves showing whether lim f(x) as x->infinity has a limit. Do you know l'Hopital's rule? For the monotone part can you show f'(x) has a constant sign for x>1? I think that's the hard part. I don't think it's a terribly easy problem.

Last edited: Jul 17, 2009
3. Jul 17, 2009

lanedance

Hi Juan Pablo

bounded from above means there exists some N such that an < N for all n

the sequence is monotonically increasing if an-an-1>0 for all n

hope this helps you get started

4. Jul 18, 2009

Juan Pablo

Thanks to you both. I helped a lot!

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